Work supported in part by US Department of Energy contract DE-AC02-76SF00515.

SLAC–PUB–13848 AEI-2009-110 ITP-UH-18/09 arXiv:0911.5704

R4 counterterm and E7(7) symmetry in maximal supergravity

Johannes Brodela,b and Lance J. Dixonc

a Max-Planck-Institut fur Gravitationsphysik

Albert-Einstein-Institut, Golm, Germany

b Institut fur Theoretische Physik

Leibniz Universitat Hannover, Germany

c SLAC National Accelerator Laboratory

Stanford University

Stanford, CA 94309, USA

Abstract

The coefficient of a potential R4 counterterm in N = 8 supergravity has been shown previously

to vanish in an explicit three-loop calculation. The R4 term respects N = 8 supersymmetry;

hence this result poses the question of whether another symmetry could be responsible for the

cancellation of the three-loop divergence. In this article we investigate possible restrictions from

the coset symmetry E7(7)/SU(8), exploring the limits as a single scalar becomes soft, as well as

a double-soft scalar limit relation derived recently by Arkani-Hamed et al. We implement these

relations for the matrix elements of the R4 term that occurs in the low-energy expansion of closed-

string tree-level amplitudes. We find that the matrix elements of R4 that we investigated all obey

the double-soft scalar limit relation, including certain non-maximally-helicity-violating six-point

amplitudes. However, the single-soft limit does not vanish for this latter set of amplitudes, which

suggests that the E7(7) symmetry is broken by the R4 term.

Contents

1 Introduction 2

2 Coset structure, hidden symmetry and double-soft limit 6

3 String theory corrections to field theory amplitudes 8

3.1 Tree-level amplitudes in N = 4 SYM and N = 8 Supergravity . . . . . . . . . . . . 9

3.2 Amplitudes in open and closed string theory . . . . . . . . . . . . . . . . . . . . . . 11

4 Setting up the calculation 14

4.1 KLT relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2 Choosing a suitable amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.3 Supersymmetric Ward identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.4 N = 1 supersymmetric Ward identities . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.5 The second N = 1 SUSY diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 E7(7) symmetry for α′-corrected amplitudes? 23

6 Conclusion 26

1

1 Introduction

Divergences of four-dimensional gravity theories have been under investigation practically

since the advent of quantum field theory. While pure gravity can be shown to be free of ul-

traviolet divergences at one loop, the addition of scalars or other particles renders the theory

nonrenormalizable [1]. At the two-loop level, the counterterm

R3 ≡ Rλρµν Rστ

λρ Rµνστ (1.1)

has been shown to respect all symmetries, to exist on-shell [2, 3] and to have a nonzero coefficient

for pure gravity [4, 5].

Supersymmetry is known to improve the ultraviolet behavior of many quantum field the-

ories. In fact, supersymmetry forbids the R3 counterterm in any supersymmetric version of

four-dimensional gravity, provided that all particles are in the same multiplet as the graviton.

That is because the operator R3 generates a scattering amplitude [6, 7] that can be shown to

vanish by supersymmetric Ward identities (SWI) [8, 9, 10, 11]. However, the next possible coun-

terterm [12, 13, 14, 15, 16] is

R4 ≡ tµ1ν1...µ4ν48 tρ1σ1...ρ4σ4

8 Rµ1ν1ρ1σ1Rµ2ν2ρ2σ2Rµ3ν3ρ3σ3Rµ4ν4ρ4σ4 , (1.2)

where t8 is defined in eq. (4.A.21) of ref. [17]. This operator, also known as the square of the

Bel-Robinson tensor [18], on dimensional grounds can appear as a counterterm at three loops.

It is compatible with supersymmetry, not just N = 1 but all the way up to maximal N = 8

supersymmetry. This property follows from the appearance of R4 in the low-energy effective action

of the N = 8 supersymmetric closed superstring [19]; indeed, it represents the first correction

term beyond the limit of N = 8 supergravity [20], appearing at order α′3. We denote by R4 the

N = 8 supersymmetric multiplet of operators containing R4.

We note that beyond the four-point level, and in more than four dimensions, it is possible

to distinguish at least one other quartic combination of Riemann tensors, maintaining N = 8

supersymmetry. In the notation of refs. [21, 22], the R4 term appearing in the tree-level closed

superstring effective action in ten dimensions is actually e−2φ(t8t8 − 18ε10ε10)R

4, where ε10 is

the ten-dimensional totally antisymmetric tensor, and φ is the (ten-dimensional) dilaton. The

dilaton is also the string loop-counting parameter, so that terms in the effective action at L loops

are proportional to exp(−2(1 − L)φ) (in string frame). The corresponding term in the one-loop

effective action in the IIA string theory differs from the IIB case in the sign of the ε10ε10 term, and

is proportional to (t8t8 + 18ε10ε10)R

4. In four dimensions, the ε10ε10 terms vanish. However, the

different possible dependences of R4 terms on the dilaton persist, and become more complicated,

because the dilaton resides in the 70 scalars of N = 8 supergravity, which are members of the 70

representation of SU(8), and the R4 prefactor should be SU(8) invariant. Green and Sethi [23]

found powerful constraints on the possible dependences in ten dimensions using supersymmetry

2

alone; indeed, only tree-level (e−2φ) and one-loop (constant) terms are allowed. It would be very

interesting to examine the analogous supersymmetry constraints in four dimensions.

The issue of possible counterterms in maximal N = 8 supergravity [24, 25] is under perpet-

ual investigation. Many of the current arguments rely on (linearized) superspace formulations

and nonrenormalization theorems [26, 27], which in turn depend on the existence of an off-shell

superspace formulation. It was a common belief for some time that a superspace formulation of

maximally-extended supersymmetric theories could be achieved employing off-shell formulations

with at most half of the supersymmetry realized. On the other hand, an off-shell harmonic su-

perspace with N = 3 supersymmetry for N = 4 super-Yang-Mills (SYM) theory was constructed

a while ago [28]. Assuming the existence of a similar description realizing six of the eight super-

symmetries of N = 8 supergravity would postpone the onset of possible counterterms at least

to the five-loop level, while realizing seven of eight would postpone it to the six-loop level [27].

However, an explicit construction of such superspace formalisms has not yet been achieved in the

gravitational case.

Another way to explore the divergence structure of N = 8 supergravity is through direct

computation of on-shell multi-loop graviton scattering amplitudes. The two-loop four-graviton

scattering amplitude [29] provided the first hints that the R4 counterterm might have a vanishing

coefficient at three loops. The full three-loop computation then demonstrated this vanishing

explicitly [30, 31]. A similar cancellation has been confirmed at four loops recently [32]. The

latter cancellation in four dimensions is not so surprising for the four-point amplitude because

operators of the form ∂2R4 can be eliminated in favor of R5 using equations of motion [33], and

it has been shown that there is no N = 8 supersymmetric completion of R5 [34, 35]. (This is

consistent with the absence of R5 terms from the closed-superstring effective action [36].) On

the other hand, the explicit multi-loop amplitudes show an even-better-than-finite ultraviolet

behavior, as good as that for N = 4 super-Yang-Mills theory, which strengthens the evidence for

an yet unexplored underlying symmetry structure.

There are also string- and M-theoretic arguments for the excellent ultraviolet behavior ob-

served to date. Using a nonrenormalization theorem developed in the pure spinor formalism for

the closed superstring [37], Green, Russo and Vanhove argued [38] that the first divergence in

N = 8 supergravity might be delayed until nine loops. (On the other hand, a very recent ana-

lysis of dualities and volume-dependence in compactified string theory by the same authors [39]

indicates a divergence at seven loops, in conflict with the previous argument.) Arguments based

on M-theory dualities suggest the possibility of finiteness to all loop orders [40, 41]. However,

the applicability of arguments based on string and M theory to N = 8 supergravity is subject to

issues related to the decoupling of massive states [42].

There have also been a variety of attempts to understand the ultraviolet behavior of N = 8

supergravity more directly at the amplitude level. The “no triangle” hypothesis [43, 44], now a

3

theorem [45, 46], states in essence that the ultraviolet behavior of N = 8 supergravity at one

loop is as good as that of N = 4 super-Yang-Mills theory. It also implies many, though not

all, of the cancellations seen at higher loops [47]. Some of the one-loop cancellations are not

just due to supersymmetry, but to other properties of gravitational theories [48], including their

non-color-ordered nature [49].

These one-loop considerations, and the work of ref. [27], suggest that conventional N = 8

supersymmetry alone may not be enough to dictate the finiteness of N = 8 supergravity. However,

since the construction of N = 8 supergravity [50, 24, 25] it has been realized that another

symmetry plays a key role — the exceptional, noncompact symmetry E7(7). Could this symmetry

contribute somehow to an explanation of the (conjectured) finiteness of the theory?

The general role of the E7(7) symmetry, regarding the finiteness of maximal supergravity,

has been a topic of constant discussion. (Aspects of its action on the Lagrangian in light-cone

gauge [51], and covariantly [52, 53], have also been considered recently.) While a manifestly

E7(7)-invariant counterterm was presented long ago at eight loops [15, 54], newer results using

the light-cone formalism cast a different light on the question [55].

In this article we investigate whether restrictions on the appearance of the R4 term could

originate directly from the exceptional symmetry. One way to test whether R4 is invariant

under E7(7) is to utilize properties of the on-shell amplitudes that R4 produces. This method is

convenient because it turns out that the amplitudes can be computed, using string theory, even

when a full nonlinear expression for R4 in four dimensions is unavailable. However, it is limited

to the matrix elements produced by the R4 term in the tree-level string effective action. As

discussed earlier, there may be other possible N = 8 supersymmetric R4 terms, distinguished for

example by their precise dependence on the scalar fields in the theory, which we will not be able

to probe in this way.

Arkani-Hamed, Cachazo and Kaplan (ACK) [46] provided a very useful tool for an amplitude-

based approach. Working in pure N = 8 supergravity, they showed recursively how generic

amplitudes with one soft scalar particle vanish as the soft momentum approaches zero. This

vanishing was first observed by Bianchi, Elvang and Freedman [56], and associated with the fact

that the scalars parametrize the coset manifold E7(7)/SU(8) and obey relations similar to soft

pion theorems [57, 58]. On the other hand, in the case of soft pion emission, the amplitude can

remain nonvanishing as the (massless) pion momentum vanishes, due to graphs in which the pion

is emitted off an external line; a divergence in the adjacent propagator cancels a power of pion

momentum in the numerator from the derivative interaction. In the supergravity case, it was

found that the external scalar emission graphs actually vanish on-shell in the soft limit [56].

ACK further considered in detail the emission of two additional soft scalar particles from a

hard scattering amplitude, and thereby derived a relation between amplitudes differing by two in

the number of legs. The relation should hold for any theory with E7(7) symmetry. If one could

4

show agreement of the single-soft limit and the ACK relation for all amplitudes derived from a

modified N = 8 supergravity action, in this case perturbing it by the R4 term, then this action

should obtain no restrictions from E7(7).

Actually, for this conclusion to hold, E7(7) should remain a good symmetry at the quantum

level. Although there is evidence in favor of this, we know of no all-orders proof. At one loop, the

cancellation of anomalies for currents from the SU(8) subgroup of E7(7) was demonstrated quite

a while ago [59]. The analysis was subtle because a Lagrangian for the vector particles cannot

be written in a manifestly SU(8)-covariant fashion. Thus the vectors contribute to anomalies,

cancelling the more-standard contributions from the fermions. More recently, the question of

whether the full E7(7) is a good quantum symmetry has been re-examined using the methods

of ACK. He and Zhu recently showed that the infrared-finite part of single-soft scalar emission

vanishes at one loop for an arbitrary number of external legs [60] as it does at tree level. (Earlier,

Kallosh, Lee and Rube [61] showed the vanishing of the four-point one-loop amplitude in the

single-soft limit for complex momenta.) A similar argument by Kaplan [62] shows that the

double-soft scalar limit relation in N = 8 supergravity can also be extended to one loop. These

results support the conjecture that the full E7(7) is a good quantum symmetry of the theory, at

least at the one-loop level.

The purpose of this article is to test the E7(7) invariance of eq. (1.2), by exploring the validity

of the single-soft limits and the ACK relation for the four-dimensional N = 8 supergravity action,

modified by adding the supersymmetric extension of the R4 term that appears in the tree-level

closed superstring effective action. The bulk of the article is devoted to the construction of

amplitudes produced by this term. As we will see, we need to go to six-point NMHV amplitudes

to get the first nontrivial result. The strategy for obtaining information about higher-order α′-

terms in closed-string scattering is the same as used in a recent article by Stieberger [36]: We

will fall back to open-string calculations [63] and derive the corresponding closed-string results

by employing the Kawai-Lewellen-Tye (KLT) [64] relations.

The remainder of this article is organized as follows. Sections 2 and 3 collect the background

information on symmetries of N = 8 supergravity, including the double-soft scalar limit of am-

plitudes, and they illuminate the state and availability of open-string amplitude calculations. In

section 4 the calculation is set up. We start by introducing the KLT relations connecting open-

and closed-string amplitudes in subsection 4.1. A suitable amplitude for probing the double-

soft scalar limit relation is singled out in subsection 4.2. The N = 1 supersymmetric Ward

identities needed to make use of the available open-string amplitudes are described in detail in

subsections 4.3, 4.4 and 4.5. The main result of this article, the testing of possible restrictions

originating from E7(7) symmetry, by employing the single- and double-soft scalar limit relations

on amplitudes produced by the R4 term, is presented in section 5. In section 6 we draw our

conclusions.

5

2 Coset structure, hidden symmetry and double-soft limit

The physical field content of the maximal supersymmetric gravitational theory in four dimen-

sions, N = 8 supergravity [24, 25], consists of a vierbein (or graviton), 8 gravitini, 28 abelian

gauge fields, 56 Majorana gauginos of either helicity, and 70 real (or 35 complex) scalars, which

can be collected together in a single massless N = 8 (on-shell) supermultiplet.

Starting from the fact that the vector bosons form an antisymmetric tensor representation

of SO(8) in the ungauged theory, Bianchi identities and equations of motion can be considered

in order to realize a much larger symmetry, which leads to the notion of generalized electric-

magnetic duality transformations. Investigating these transformations more closely and enlarging

the corresponding duality group maximally by adding further scalars, not all of which turn out

to be physical. After gauging a resulting local SU(8) symmetry in order to reduce the degrees of

freedom of the generalized duality group, 70 physical scalars remain. These scalars parameterize

the cosetE7(7)

SU(8) [25, 65], where E7(7) denotes a noncompact real form of E7, which has SU(8) as

its maximal compact subgroup. In other words, the scalars can be identified with the noncompact

generators of E7(7). The resulting gauge is called unitary.

More explicitly, in unitary gauge the 63 compact generators T JI of SU(8) can be joined with

70 generators XI1...I4 to form the adjoint representation of E7(7). Here XI1...I4 transforms under

SU(8) in the four-index antisymmetric tensor representation (I, J = 1, . . . , 8). The commutation

relations between those generators are given schematically by

[T, T ] ∼ T , [X,T ] ∼ X , and [X,X] ∼ T . (2.1)

The first commutator is just the usual SU(8) Lie algebra, and the second one follows straightfor-

wardly from the identification of X with the 70 of SU(8). The more nontrivial statement about

E7(7) invariance resides in the third commutator in eq. (2.1). Assuming the two scalars to be

represented as XI1...I41 and X2 I5...I8 , where the upper-index version can be obtained by employing

the SU(8)-invariant tensor,

XI1...I4 =1

24εI1I2I3I4I5I6I7I8XI5...I8 , (2.2)

the third relation reads explicitly (see e.g. ref. [46]),

−i [XI1...I41 ,X2 I5...I8] = εJI2I3I4

I5I6I7I8T I1

J + εI1JI3I4I5I6I7I8

T I1J + . . . + εI1I2I3I4

I5I6I7J T JI8 . (2.3)

Here εI1I2I3I4I5I6I7I8

= 1,−1, 0 if the upper index set is an even, odd or no permutation of the lower set,

respectively. (For a more general discussion of the properties of E7(7), see appendix B of ref. [25].)

Amplitudes in N = 8 supergravity are invariant under SU(8) rotations by construction. On

the other hand, the action of the coset symmetryE7(7)

SU(8) on amplitudes is not obvious. One

can understand the connection by recalling that the vacuum state of the theory is specified by

6

the expectation values of the physical scalars. Because the scalars are Goldstone bosons, the

soft emission of scalars in an amplitude changes the expectation value and moves the theory to

another point in the vacuum manifold.

Arkani-Hamed, Cachazo and Kaplan [46] used the BCFW recursion relations [66, 67] to in-

vestigate how the noncompact part of E7(7) symmetry controls the soft emission of scalars in

N = 8 supergravity. Consider first the emission of a single soft scalar (which was also studied in

refs. [56, 68]). The corresponding amplitudes can be traced back via the BCFW recursion rela-

tions to the three-particle amplitude, whose vanishing in the soft limit can be shown explicitly.

Hence the emission of a single scalar from any amplitude vanishes in N = 8 supergravity,

Mn+1(1, 2, . . . , n + 1) −−−→p1→0

0 , (2.4)

where p1 denotes the vanishing scalar momentum.

Moving on to double-soft emission, several different situations have to be distinguished, which

are labelled by the number of common indices between the sets I1, I2, I3, I4 and I5, I6, I7, I8in eq. (2.3). Four common indices allow the creation of an SU(8) singlet, corresponding to the

emission of a single soft graviton. This case is not interesting because [X,X] vanishes. Similarly,

if the scalars share one or two indices, the situation corresponds to a single soft limit in one

of the subamplitudes generated by the BCFW recursion relations; thus this limit vanishes, and

does not probe the commutator in eq. (2.3). Another way to see the vanishing is to reconsider

eq. (2.3) explicitly: there are simply not enough indices to saturate the right-hand side. The

only interesting configuration occurs if the two scalars X1 and X2 agree on exactly three of

their indices. This result is in accordance with the commutation relation eq. (2.3), where three

equal indices are necessary for the commutator of two noncompact generators to yield a result

proportional to an SU(8) generator.

Performing an explicit calculation of an (n + 2)-point supergravity tree amplitude Mn+2

containing two scalars sharing three indices and considering the double-soft limit on X1 and X2

results in the double-soft limit [46]

Mn+2(1, 2, . . . , n + 2) −−−−−→p1,p2→0

1

2

n+2∑

i=3

pi · (p2 − p1)

pi · (p1 + p2)T (ηi)Mn(3, 4, . . . , n + 2) , (2.5)

where

T (ηi)JK = T

(

[XI1...I4,XI5...I8])J

K= εI1I2I3I4K

I5I6I7I8J × ηiK∂ηiJ(2.6)

acts on (Mn)KJ ; the n-point amplitude Mn has open SU(8) indices due to the particular choice

of indices of the scalars. Again, εI1I2I3I4KI5I6I7I8J = 1,−1, 0 if the upper index set is an even, odd or no

permutation of the lower set.

The Grassmann variables ηiA in the argument of eq. (2.6) refer to the description of an

amplitude in the so-called on-shell superspace formalism [69]. They are a set of 8n anticommuting

7

objects, where the index i = 1, . . . , n numbers the particles and A is an SU(8) index. Using these

variables, one can write down a generating functional for MHV amplitudes in supergravity [56],

Ωn =1

256

Mn(B−1 , B−

2 , B+3 , B+

4 , . . . , B+n )

〈12〉88

∏

A=1

n∑

i,j=1

〈ij〉ηiAηjA , (2.7)

where B± are positive and negative helicity gravitons. Particle states of the N = 8 multiplet can

be identified with derivatives with respect to the anticommuting variables

1 ↔ B+i

∂

∂ηiA↔ FA+

i · · · ∂4

∂ηiA∂ηiB∂ηiC∂ηiD↔ XABCD · · ·

· · · − 1

7!εABCDEFGH

∂7

∂ηiB∂ηiC . . . ∂ηiH↔ F−

iA · · · 1

8!εABCDEFGH

∂8

∂ηiA∂ηiB . . . ∂ηiH↔ B−

i ,

(2.8)

where the number of η’s is connected to the helicity of the state, and F± denote gravitini of

either helicity. Acting with these operators on the generating functional (2.7), one obtains the

correct expressions for the corresponding component amplitudes, which automatically obey the

MHV supersymmetry Ward identities. For example a two-gravitino two-graviton amplitude will

read:

〈F 5+ F−5 B+ B−〉 ≡ M4(F

5+1 , F−

2,5, B+3 , B−

4 )

= −(

∂

∂η15

)(

1

7!ε12345678

∂7

∂η21 . . . ∂η24∂η26 . . . ∂η28

)

×(

1

8!ε12345678

∂8

∂η41 . . . ∂η48

)

Ω4 . (2.9)

As we will see below, the SU(8) generator (2.6) will act consistently on the remnant of the

six-point amplitude represented in the above formalism.

In the double-soft limit (2.5), the amplitude with two soft scalars sharing three indices becomes

a sum of amplitudes with only hard momenta; in each summand one leg gets SU(8) rotated by

an amount depending on its momentum. This relation has been proven by ACK at tree-level for

pure N = 8 supergravity. Here we will construct a suitable α′-corrected amplitude, derived from

an action containing the supersymmetrized version of the R4 term, and then take the double-soft

limit numerically in order to test the E7(7) invariance of this term.

In order to do so, we will first have a look at string theory corrections to field theory amplitudes

in the next section, before we set up the actual calculation in section 4.

3 String theory corrections to field theory amplitudes

Tree amplitudes for Type I open and Type II closed string theory have been computed and

expanded in α′ for various collections of external states. The leading terms in the low-energy

8

effective action are N = 4 SYM and N = 8 supergravity, respectively [20]. Indeed, in the zero

Regge slope limit (α′ → 0), the string amplitudes agree with the corresponding field theory

results.

Expanding the string theory amplitude further in α′ yields corrections to the field-theoretical

expressions, which can be summarized by a series of local operators in the effective field theory.

Terms which have to be added to the N = 4 SYM and N = 8 supergravity actions in order

to reproduce the α′ corrections have been identified for low orders in α′. In particular, the first

nonzero string correction to the action of N = 8 supergravity is the supersymmetrized version of

the possible R4 counterterm eq. (1.2) discussed above [19].

The next subsection reviews properties of amplitudes in maximally supersymmetric field the-

ories. Some recent computations of string theory amplitudes and their low-energy expansions are

discussed in the following subsection.

3.1 Tree-level amplitudes in N = 4 SYM and N = 8 Supergravity

A general amplitude in N = 4 SYM can be color-decomposed as

ASYMn (1, 2, . . . , n) = gn−2

YM

∑

σ∈Sn/Zn

Tr(T aσ(1) · · ·T aσ(n))ASYMn (σ(1), σ(2), . . . , σ(n)), (3.1)

where the summation is over all (n − 1)! non-cyclic permutations of i = 1, 2, . . . , n. The number

i is understood as a collective label for the momentum pi and helicity hi of particle i, e.g.

1 ≡ (p1, h1), and the T ai are matrices in the fundamental representation of the Yang-Mills gauge

group SU(Nc), normalized to Tr(T aT b) = δab.

The gauge-invariant subamplitudes ASYMn are independent of the color structure and can be

shown to exhibit the following properties [70]:

• invariance under cyclic permutations: ASYMn (1, 2, . . . , n) = ASYM

n (2, 3, . . . , n, 1)

• reflection identity: ASYMn (1, 2, . . . , n) = (−1)nASYM

n (n, n − 1, . . . , 2, 1)

• photon decoupling (or dual Ward) identity:

ASYMn (1, 2, 3, . . . , n) + ASYM

n (2, 1, 3, . . . , n) + ASYMn (2, 3, 1, . . . , n)

+ · · · + ASYMn (2, 3, . . . , 1, n) = 0. (3.2)

In addition, amplitudes in maximally supersymmetric theories are classified by their helicity

structure. Employing supersymmetric Ward identities (see section 4.3), pure-gluon amplitudes

with helicity structure (± + · · ·+) can be shown to vanish [8, 9]. The simplest nonvanishing

9

configurations (− − + · · ·+) are called maximally helicity violating (MHV) amplitudes. In the

case that all external legs are gluons g±, they are given by [71]:

ASYMn (g−1 , g−2 , g+

3 , . . . , g+n ) = i

〈12〉4〈12〉〈23〉 · · · 〈n1〉 , (3.3)

where |k±i 〉 are massless Weyl spinors, normalized by 〈ij〉[ji] = 2ki · kj , and 〈k−

i |k+j 〉 = 〈ij〉,

〈k+i |k−

j 〉 = [ij]. The simplicity of the MHV sector is also expressed in the relations between

different MHV amplitudes: any MHV amplitude is related directly to the pure-gluon one by

supersymmetric Ward identities (see section 4.3), so that the knowledge of eq. (3.3) determines

the complete set of MHV amplitudes.

While in the four- and five-point case the only nonvanishing configurations are MHV (or

anti-MHV), the advent of a sixth leg introduces a new class of helicity structures, the so-called

next-to-MHV (NMHV) amplitudes. Here it is necessary to distinguish three different helicity

orderings

X : (−−− + ++) Y : (−− + − ++) Z : (− + − + −+) . (3.4)

Expressions for the amplitudes are distinct for the different orderings X, Y and Z. However,

there is no procedural difference in deriving the expressions, so we will generally illustrate the

amplitudes and supersymmetry relations for the helicity configuration X. Explicit results for all

six-point pure-gluon NMHV amplitudes can be found in ref. [70], for example. More compact

expressions result from use of the BCFW recursion relations. Using these relations, a prescription

for determining all tree-level amplitudes in N = 4 SYM from superconformal invariants has been

derived [72].

We note that the supersymmetric Ward identities, reflection symmetry and cyclic invariance

— as well as parity, or spinor conjugation — relate amplitudes within a certain NMHV helicity

ordering only (X, Y or Z). On the other hand, the photon decoupling identity is an example of

a relation among amplitudes featuring different helicity orderings.

Next we turn to amplitudes in N = 8 supergravity. In this case, the color trace, which

forces particles in gauge-theory subamplitudes to remain in a certain cyclic order, does not exist.

Instead, supergravity amplitudes are symmetric under exchange of particles with the same helicity.

We write the full amplitude MSUGRAn (1, 2, . . . , n) as

MSUGRAn (1, 2, . . . , n) =

(κ

2

)(n−2)MSUGRA

n (1, 2, . . . , n), (3.5)

where only the gravitational coupling constant κ =√

32πGN has been removed from MSUGRAn .

The four- and five-point MHV amplitudes for gravitons B± are given by [73]

MSUGRA4 (B−

1 , B−2 , B+

3 , B+4 ) = i 〈12〉8 [12]

〈34〉N(4),

MSUGRA5 (B−

1 , B−2 , B+

3 , B+4 , B+

5 ) = i 〈12〉8 ε(1, 2, 3, 4)

N(5),

(3.6)

10

where

ε(i, j,m, n) = 4iεµνρσkµi kν

j kρmkσ

n = [ij]〈jm〉[mn]〈ni〉 − 〈ij〉[jm]〈mn〉[ni] (3.7)

and

N(n) ≡n−1∏

i=1

n∏

j=i+1

〈ij〉 . (3.8)

The higher-point MHV graviton amplitudes were first written down in ref. [73]. Explicit expres-

sions for other helicity configurations are rare. However, in ref. [74] a prescription is given how

to calculate any N = 8 supergravity tree-level amplitude by employing “gravity subamplitudes”,

BCFW recursion relations, and superconformal invariants [75, 72].

In the on-shell superspace formalism introduced above, the determination of the type of

amplitude away from those containing gluons (gravitons) exclusively can be done by counting

derivatives acting on the appropriate generating functional. While 8 (16) derivatives are necessary

for MHV amplitudes in N = 4 SYM (N = 8 supergravity), there are 12 (24) derivatives in the

NMHV sector.

3.2 Amplitudes in open and closed string theory

Open-string tree amplitudes An have the same color decomposition (3.1), with ASYMn replaced

by the color-ordered string subamplitude An. At the four-point level, the two subamplitudes are

related by the Veneziano formula,

A4(1−, 2−, 3+, 4+) = V (4)(s1, s2)ASYM

4 (1−, 2−, 3+, 4+)

=Γ(1 + s1)Γ(1 + s2)

Γ(1 + s1 + s2)ASYM

4 (1−, 2−, 3+, 4+) . (3.9)

The above expression is given in terms of kinematical invariants defined via

[[i]]n = α′ (ki + ki+1 + · · · + ki+n)2 , sj = sj j+1 = [[j]]1 , tj = [[j]]2 , (3.10)

which are s1 = [[1]]1 = s12 = 2α′k1 · k2 and s2 = [[2]]1 = s23 = 2α′k2 · k3 on-shell. Expanding the

form-factor V (4) in powers of α′ one finds

V (4)(s1, s2) = 1 − ζ(2)s1s2 + ζ(3)s1s2(s1 + s2) + O(α′4), (3.11)

where the leading correction to the pure Yang-Mills amplitude arises from the interaction term

of four gauge field-strength tensors [76, 17, 77].

The full open string amplitude is quite simple in the four-point case (3.9). On the other

hand, its generalizations to more external legs turn out to involve generalized hypergeometric

11

functions [78]. Any n-point open string amplitude can be expressed in terms of (n − 3)! hyper-

geometric basis integrals. Expanding those functions in powers of α′ yields expressions for the

string-corrected five- and six-point MHV amplitudes

A5 =

[

V (5)(sj) −i α′2

2ε(1, 2, 3, 4)P (5)(sj)

]

ASYM5

A6 =

[

V open6 (sj , tj) −

i α′2

2

5∑

k=1

εkP(6)k (sj, tj)

]

ASYM6 , (3.12)

where

ε1 = ε(2, 3, 4, 5), ε2 = ε(1, 3, 4, 5), ε3 = ε(1, 2, 4, 5), ε4 = ε(1, 2, 3, 5), ε5 = ε(1, 2, 3, 4) .

(3.13)

Expansions in α′ are given by [79]

V (5)(si) = 1 − ζ(2)

2(s1s2 + s2s3 + s3s4 + s4s5 + s5s1)

+ζ(3)

2

(

s21s2 + s2

2s3 + s23s4 + s2

4s5 + s25s1 + s1s

22 + s2s

23 + s3s

24 + s4s

25 + s5s

21

+ s1s3s5 + s2s4s1 + s3s5s2 + s4s1s3 + s5s2s4

)

+ O(α′4) , (3.14)

P (5)(si) = ζ(2) − ζ(3)(s1 + s2 + s3 + s4 + s5) + O(α′2) , (3.15)

and explicit expressions for V (6) and P(6)k can be found in the same reference.

Stieberger and Taylor have pushed the calculations even further [63]. In the process of de-

termining all pure-gluon NMHV six-point amplitudes, they computed the following additional

auxiliary amplitudes for the helicity configuration X defined in eq. (3.4):

〈φ−φ−φ−φ+φ+φ+〉 , 〈φ−φ−λ−λ+φ+φ+〉 , and 〈φ−φ−g−g+φ+φ+〉 , (3.16)

as well the analogous quantities for Y and Z. Here λ denotes a gluino and φ a scalar. In order

to get an impression of the complexity of the result, we provide the pure-gluon NMHV six-

point amplitude in helicity configuration X [63], which will be expressed employing the following

kinematic variables:

αX = − [12]〈34〉[ 6|X|5〉 , βX = [12]〈45〉[ 6|X|3〉 , γX = [61]〈34〉[2|X|5〉 , (3.17)

where X ≡ p6 + p1 + p2. The subamplitude reads1

A6(g+1 , g+

2 , g−3 , g−4 , g−5 , g+6 ) =

α′5

s5

(

NX1

α2X

s21s

23

+ NX2

β2X

s21

+ NX3

γ2X

s23

+ NX4

αXβX

s21s3

+ NX5

αXγX

s1s23

+ NX6

βXγX

s1s3

)

, (3.18)

1Note the shifted ordering of helicities compared to eq. (3.4). A cyclic shift (1, 2, 3, 4, 5, 6) → (3, 4, 5, 6, 1, 2) has

to be performed in order to match the results analytically with ref. [63].

12

where the expansion of the functions NX to O(α′2) is:

NX1 = −ζ(2) s1s3 + . . . ,

NX2 =

s1

s2s4t1− ζ(2)

(

s1s6

s2s4+

s21

s4t1+

s1s5

s2t1

)

+ . . . ,

NX3 =

s3

s2s6t2− ζ(2)

(

s3s4

s2s6+

s3s5

s2t2+

s23

s6t2

)

+ . . . ,

NX4 = ζ(2)

(

s1t2s2

+s1t3s4

)

+ . . . ,

NX5 = ζ(2)

(

s3t1s2

+s3t3s6

)

+ . . . ,

NX6 =

t3s2s4s6

+ ζ(2)

(

s1 + s3 − s5

s2− t1t3

s2s4− t2t3

s2s6− t23

s4s6

)

+ . . . . (3.19)

The low-energy limit of closed Type II string theory in four dimensions is N = 8 supergravity.

The first correction to the low-energy effective action can be determined from the expression for

the closed string four-point amplitude, or Virasoro-Shapiro amplitude,

M4(1−, 2−, 3+, 4+) = V

(4)closed(s1, s2)MSUGRA

4 (1−, 2−, 3+, 4+)

=Γ(1 + s1)Γ(1 + s2)Γ(1 − s1 − s2)

Γ(1 − s1)Γ(1 − s2)Γ(1 + s1 + s2)MSUGRA

4 (1−, 2−, 3+, 4+) . (3.20)

The expansion of V(4)closed has the first nonvanishing correction at O(α′3),

V(4)closed(s1, s2) = 1 + 2 ζ(3) s1s2(s1 + s2) + O(α′4) , (3.21)

which corresponds to a supersymmetrized version of eq. (1.2) in the low energy effective action [19].

In other words, keeping terms up to order O(α′3) in the closed-string amplitudes is equivalent to

working with a theory whose effective action is given by

Scorr =

∫

d4x√−g(R + α′3R4) + O(α′4) . (3.22)

While α′-corrected six-point amplitudes in open string theory (N = 4 SYM) are already very

cumbersome to calculate, the situation is even worse for closed string theory (N = 8 super-

gravity). For higher-point tree amplitudes it is therefore more convenient to rely on the KLT

relations, which express closed string amplitudes as simple quadratic combinations of open string

amplitudes.

Several different cyclic orderings of the open string amplitudes are required as input to the

KLT relations. Fortunately, there are several open string amplitudes available. In particular, a

couple of six-point NMHV amplitudes have been computed [63]2, which will serve below as input

to the calculation of a suitable α′-corrected N = 8 supergravity amplitude.

2We are grateful to Stephan Stieberger and Tomasz Taylor for providing us with expressions for the amplitudes

from ref. [63] through order α′3.

13

4 Setting up the calculation

Arkani Hamed, Cachazo and Kaplan have proven eq. (2.5) analytically, by employing BCFW

recursion relations for N = 8 supergravity with E7(7) realized on-shell. Because invariance under

E7(7) is a necessary condition for the relation to be valid, eq. (2.5) provides a useful tool for

testing other theories, or operators, for their symmetry properties under E7(7). In particular, if

the double-soft limit of all (n + 2)-point amplitudes derived from eq. (3.22) coincides with the

SU(8) rotated sum of the corresponding n-point amplitudes, that would be strong evidence that

E7(7) symmetry does not restrict the appearance of R4 as a counterterm in N = 8 supergravity.

The analytical approach that ACK used to prove eq. (2.5) does not hold for the α′-corrected

N = 8 amplitudes. Higher-dimension operators lead to poorer large-momentum behavior, so

that amplitudes shifted by large complex momenta will not fall off fast enough for the BCFW

recursion relations to be valid. Instead we will find explicit (if lengthy) expressions for suitable

and available string theory amplitudes, from which the α′-corrected amplitudes corresponding to

eq. (3.22) can be deduced, and their double-soft limits inspected (numerically).

After we give a short introduction to the KLT relations in subsection 4.1, we will explore

the constraints on the α′-corrected N = 8 supergravity amplitude originating from the double-

soft limit relation eq. (2.5) in subsection 4.2. Appropriate N = 8 amplitudes will be identified

and decomposed into N = 4 SYM matrix elements using the KLT relations. The required (α′-

corrected) N = 4 SYM matrix elements can be related to the available open string amplitudes

by carefully examining the NMHV supersymmetric Ward identities. In subsections 4.3 and 4.4,

the N = 1 supersymmetric Ward identities will be reviewed in detail and used to finally obtain

expressions for the N = 4 amplitudes, which serve as input to the KLT relations, in section 5.

4.1 KLT relations

Tree-level amplitudes in closed and open string theories are linked by the KLT relations [64],

which arise from the fact that any closed-string vertex operator can be represented as a product

of two open-string vertex operators,

V closed(zi, zi) = V openleft (zi)V

openright(zi) . (4.1)

While in the closed-string amplitude the insertion points zi, zi of vertex operators are integrated

over a two-sphere, in the open-string case the real zi are integrated over the boundary of a

disk. Thus the closed-string integrand equals the product of two open-string integrands. KLT

related the two sets of string amplitudes by evaluating the closed-string integrals via a contour

deformation in terms of the open-string integrals.

14

The KLT relations for four-, five- and six-point amplitudes are

M4(1, 2, 3, 4) =−i

α′πsin(πs12)A4(1, 2, 3, 4)A4(1, 2, 4, 3) , (4.2)

M5(1, 2, 3, 4, 5) =i

α′2π2sin(πs12) sin(πs34)A5(1, 2, 3, 4, 5)A5(2, 1, 4, 3, 5)

+ P(2, 3) , (4.3)

M6(1, 2, 3, 4, 5, 6) =−i

α′3π3sin(πs12) sin(πs45)A6(1, 2, 3, 4, 5, 6)

× [sin(πs35)A6(2, 1, 5, 3, 4, 6) + sin(π(s34 + s35))A6(2, 1, 5, 4, 3, 6)]

+ P(2, 3, 4) . (4.4)

Formulae for higher-point amplitudes can be derived straightforwardly [64]. In the field-theory

(α′ → 0) limit, a closed form has been obtained for all n [43].

The above equalities are exact relations between string theory amplitudes, and so they are

valid order by order in α′. In order to calculate the string correction to an N = 8 supergravity

amplitude at a certain order in α′ from known α′-corrected expressions in N = 4 SYM, one has to

determine all combinations of terms from the expansions of the amplitudes and the sine functions,

whose multiplication results in the correct power of α′. For instance the second-order correction

to the five-point amplitude in supergravity corresponds to terms of O(α′4), due to the prefactor

of 1α′2 . Taking the absence of first-order corrections to N = 4 SYM amplitudes into account, four

combinations have to be considered in eq. (4.3), according to the following table:

sin(πs12) sin(πs34) A5(1, 2, 3, 4, 5) A5(2, 1, 4, 3, 5)

O(α′1) O(α′1) O(α′0) O(α′2)

O(α′1) O(α′1) O(α′2) O(α′0)

O(α′3) O(α′1) O(α′0) O(α′0)

O(α′1) O(α′3) O(α′0) O(α′0)

yielding

MO(α′2)5 =

is12s34

α′2

[

ASYM5 (1, 2, 3, 4, 5)A

O(α′2 )5 (2, 1, 4, 3, 5) + A

O(α′2)5 (1, 2, 3, 4, 5)ASYM

5 (2, 1, 4, 3, 5)

−π2

6 (s212 + s2

34)ASYM5 (1, 2, 3, 4, 5)ASYM

5 (2, 1, 4, 3, 5)]

+ P(2, 3) .

(4.5)

The above expression can be shown to vanish analytically, in accordance with the higher-point

generalization of eq. (3.21), or alternatively eq. (3.22), the statement that the first correction to

the closed-string effective action is at O(α′3).

Although the KLT relations are often applied to pure-graviton and pure-gluon amplitudes,

their use is not limited to these scenarios. Any pair of consistent open-string amplitudes is related

15

to an amplitude in closed string theory and vice versa. Considering the combination of two open-

string vertex operators into a closed one in eq. (4.1), one can immediately determine which type

of particle has to appear at a certain position on the supergravity side by adding the helicities

and combining the indices, according to the tensor-product decomposition of the Fock space,

[N = 8] ↔ [N = 4]L ⊗ [N = 4]R . (4.6)

Somewhat remarkably, the opposite statement is true as well: given a certain operator, corre-

sponding to a particular state in N = 8 supergravity, the helicity, global symmetry properties,

and the consistent action of supercharges in either of the theories are sufficient to unambiguously

determine the decomposition into N = 4 SYM states [56]. The decompositions relevant for the

calculation to follow are

B+ = g+g+ , F a+ = λa+g+ , F r+ = g+λr+ ,

B− = g−g− , F−a = λ−

a g− , F−r = g−λ−

r ,

Xabcd = εabcd g− g+ , Xabcr = εabcd λ−d λr+ , Xabrs = φab φrs ,

Xabcd = εabcd g+ g− , Xabcr = εabcd λd+ λ−r , Xabrs = φab φrs , (4.7)

where capital letters B, F, X denote the graviton, gravitino and scalar particle in N = 8 super-

gravity and g, λ, φ the gluon, gluino and scalar in N = 4 SYM. Quantities with indices a, b, . . .

correspond to the first SU(4), while quantities with a tilde and indices r, s, . . . are in the second

SU(4). (In particular, g does not denote a gluino!) Finally, the superscripts + and − mark the

helicity signature.

4.2 Choosing a suitable amplitude

The simplest scenario one might think of, in order to test the double-soft scalar limit relation (2.5),

would be to start with a five-point amplitude, which in turn would lead to a sum of three-point

amplitudes on the right-hand side of the relation. Three-point amplitudes are special as they

require a setup with complex momenta in order to be non-trivial. However, here we have to take

another constraint into account: we want to test amplitudes that receive nonvanishing corrections

from the R4 term. Because the interactions originating in this counterterm candidate start at

the four-point level, it is not sufficient to consider three-point amplitudes.

Therefore we will have to consider a six-point amplitude, which should reduce to a sum of four-

point amplitudes in the double-soft limit. We again require that the four-point amplitudes on the

right-hand side of eq. (2.5) are nonvanishing, which implies that they are MHV (or equivalently

anti-MHV). Fortunately, corrections to all MHV-amplitudes with four legs are known up to

O(α′3), indeed to arbitrary orders in α′, using eq. (3.20) and the MHV supersymmetry Ward

identities.

16

On the left-hand side of eq. (2.5) the situation is more intricate. The four particles that

appear already on the right-hand side are now accompanied by two additional scalars. According

to eq. (2.8), the number of η derivatives acting on the generating functional is increased by eight,

four for each scalar, so that the resulting amplitude resides in the NMHV sector. In addition,

the two scalars have to share three SU(8) indices, as elaborated on in section 2. Sorting out

the distribution of the scalars’ indices into two SU(4) subgroups, there are finally five possible

distinct choices3 satisfying the constraints. They are listed here, together with their respective

KLT decompositions according to equation (4.7):

〈XabrsXabrt · · · · 〉 → 〈φab φab · · · · 〉L × 〈φrs φrt · · · · 〉R, (4.8)

〈XabrcXabrs · · · · 〉 → 〈εabcdλ−d φab · · · · 〉L × 〈λr+ φrs · · · · 〉R, (4.9)

〈XabrcXabrd · · · · 〉 → 〈λ−d λc+ · · · · 〉L × 〈λr+ λ−

r · · · · 〉R, (4.10)

〈XabcrXabcs · · · · 〉 → 〈λ−d λd+ · · · · 〉L × 〈λr+ λ−

s · · · · 〉R, (4.11)

〈XabcdXabcr · · · · 〉 → 〈g− λd+ · · · · 〉L × 〈g+ λ−r · · · · 〉R . (4.12)

Here the ellipses are understood to be filled with four particles such that the L- and R-amplitudes

on the right-hand side of the KLT relation each transform as an SU(4) singlet. In each of

equations (4.10) to (4.12) we have left out a factor of εabcdεabcd. Because these indices are not

summed over, this factor is equal to unity. Note that 〈XabcdXabce · · · · 〉 is absent because the

five SU(4) indices a, b, c, d, e cannot be made all distinct.

In order to proceed, we need to use supersymmetric Ward identities to relate one of the

five decompositions (4.8)–(4.12) to the available open-string six-point results (see eq. (3.16) in

section 3):

〈g− g− g− g+ g+ g+〉 , 〈φ− φ− φ− φ+ φ+ φ+〉,〈φ− φ− λ− λ+ φ+ φ+〉 and 〈φ− φ− g− g+ φ+ φ+〉 . (4.13)

Supersymmetric Ward identities can be classified by the amount of supersymmetry employed (e.g.,

N = 1, 2, 4), as well as the number of legs and the sector (MHV, NMHV, etc.) characterizing

the amplitudes. We deal with six-point NMHV amplitudes exclusively here. The notation N = 4

SWI will refer to the set of supersymmetric Ward identities relating six-point NMHV amplitudes

built from the full N = 4 multiplet (g±, λ±m, φ±

n ), where m = 1, 2, 3, 4 and n = 1, 2, 3. (Note that

a superscript ± on φ implies a complex field with a different index labelling from the real φab used

above.) In the original article [63], N = 2 supersymmetric Ward identities have been served to

relate the latter three amplitudes in eq. (4.13) to the pure-gluon one. So the obvious idea would

be to search in the decompositions (4.8)–(4.12) for one in which the amplitudes contain particles

from a single N = 2 multiplet (plus its CPT conjugate), (g±, λ±m, φ±

1 ) with m = 1, 2.

3Another five combinations can be obtained by switching the left and right SU(4).

17

However, the third amplitude in eq. (4.13) contains only one type of fermion, which points into

the direction of a N = 1 multiplet. Setting up the calculation employing N = 1 SWI exclusively

is a bit simpler than using N = 2 SWI: For six-point NMHV amplitudes an explicit and simple

solution to the N = 1 SWI is known [56, 9]. (We note that very recently the supersymmetric

Ward identities in maximally supersymmetric N = 4 super-Yang-Mills theory and N = 8 super-

gravity were solved, quite remarkably, for arbitrary n-point NkMHV amplitudes [80] in terms of

basis amplitudes, in a manifestly supersymmetric form. These results may prove very useful in

extending the considerations of this paper to greater numbers of legs.)

Now the decompositions (4.8) to (4.12) are not all equally suited to the use of an N = 1 SWI.

For example, the left SU(4) amplitude of eq. (4.9) contains three distinct SU(4) indices, a, b, d,

thus requiring a full N = 4 multiplet. The other four decompositions contain amplitudes which

can be constructed from SWI with less supersymmetry. Indeed, the decomposition (4.12) contains

only one index for the left SU(4) amplitude, and one for the right one; this decomposition is the

one we will use in this paper. As will be explained below, it is possible to obtain everything we

need for testing the double-soft limit through eq. (4.12), by using a two-step procedure employing

two different sets of N = 1 SWI based on the multiplets (g±, λ±) and (φ±, λ±).

The next three subsections introduce the SWI in general, elaborate on the N = 1 SWI for

(g±, λ±) in particular, and then describe the analogous set of N = 1 SWI for the multiplet

(φ±, λ±). Then, in section 5, we will assemble these ingredients in order to test the E7(7) sym-

metry.

4.3 Supersymmetric Ward identities

Supersymmetric Ward identities can be derived using the fact that supercharges annihilate the

vacuum of the theory, Q|0〉 = 0, so that

0 = 〈[Q,β1β2 · · · βn]〉 =

n∑

i=1

〈β1β2 · · · [Q,βi] · · · βn〉 . (4.14)

Here the βi are arbitrary states from the multiplet under consideration, Q = Q(η) = 〈Qη〉 is a

supersymmetry operator, which has been bosonized by contraction with the Grassmann variable

η, and 〈β1β2 · · · βn〉 will be called the source term for the SWI. Source terms need to have an

odd number of fermions, because amplitudes derived by acting on terms with an even number

of fermions will vanish trivially. An immediate and standard result implied by eq. (4.14) is the

disappearance of all amplitudes with helicity structure 〈+++ · · ·+〉 and 〈+−− · · · −〉. With only

little more effort one can show that maximally helicity violating amplitudes (MHV) are related

pairwise by SWI, which in turn means that knowing one amplitude determines the whole MHV

sector for a particular number of legs [81]. In the NMHV sector this is no longer true; here each

18

supersymmetric Ward identity relates three amplitudes, which requires two known amplitudes in

order to determine a third one.

Stieberger and Taylor have explicitly proven for open string theory on the disk that the forms

of the supersymmetric Ward identities to all orders in α′ are identical to those in the corresponding

four-dimensional field-theoretical limit [82]. So the exploration in the next two subsections will

be valid as well for the α′-corrected amplitudes under investigation.

4.4 N = 1 supersymmetric Ward identities

As an example, let us investigate the set of amplitudes involving gluons (g+, g−) and a single pair

of gluinos (λ+, λ−) (from which we drop the SU(4) index for simplicity). The states are related

by N = 1 supersymmetry via[

Q(η), g+(p)]

= [pη] λ+(p),[

Q(η), λ+(p)]

= −〈pη〉g+(p),[

Q(η), g−(p)]

= 〈pη〉λ−(p),[

Q(η), λ−(p)]

= − [pη] g−(p), (4.15)

where Q(η) = 〈Qη〉.For each NMHV helicity sector, there are 20 distinct amplitudes related by N = 1 SWI:

a pure-gluon amplitude, a pure-gluino amplitude, nine two-gluino four-gluon amplitudes, and

nine four-gluino two-gluon amplitudes, as shown in figure 1. In the following, we assume that

amplitudes are drawn from helicity configuration X in eq. (3.4). For the two other configurations

Y and Z, the relations are completely analogous.

〈g− g− g− g+ g+ g+〉

〈λ− g− g− λ+ g+ g+〉 · · · 〈g− λ− g− g+ λ+ g+〉 · · · 〈g− g− λ− g+ g+ λ+〉

〈λ− λ− g− λ+ λ+ g+〉 · · · 〈λ− g− λ− λ+ g+ λ+〉 · · · 〈g− λ− λ− g+ λ+ λ+〉

〈λ− λ− λ− λ+ λ+ λ+〉

Figure 1: Amplitudes related by N = 1 supersymmetric Ward identities.

Amplitudes in adjacent rows of figure 1 are related by the N = 1 SWI. Acting for example

with the supersymmetry operator Q(η) on the source term 〈g− g− g− λ+ g+ g+〉 yields

〈4η〉〈g− g− g− g+ g+ g+〉 − 〈1η〉〈λ− g− g− λ+ g+ g+〉− 〈2η〉〈g− λ− g− λ+ g+ g+〉 − 〈3η〉〈g− g− λ− λ+ g+ g+〉 = 0 , (4.16)

19

which relates the pure-gluon amplitude to the two-gluino four-gluon ones from the second row

in figure 1. Due to the freedom in choosing the two-component supersymmetry parameter η,

the result is a system of equations which has rank 2. In order to find all relations between the

pure gluon amplitude (first row) and the amplitudes in the second row, the action of Q(η) on all

possible source terms featuring one gluino and five gluons,

〈λ− g− g− g+ g+ g+〉, 〈g− λ− g− g+ g+ g+〉, 〈g− g− λ− g+ g+ g+〉,〈g− g− g− λ+ g+ g+〉, 〈g− g− g− g+ λ+ g+〉, 〈g− g− g− g+ g+ λ+〉, (4.17)

has to be considered. The resulting system, linking ten amplitudes from the first and second

rows, turns out to have rank eight, thus requiring two known amplitudes in order to derive all

the others.

Repeating the analysis for the second and third rows, there are notably more identities to

consider. They are generated by acting with Q(η) on any of the 18 different source terms built

from three gluinos and the same number of gluons, e.g. 〈λ− λ− g− g+ λ+ g+〉. Interestingly this

system connecting 18 unknown amplitudes is of rank 16, meaning that again two amplitudes have

to be known in order to fix all the others.

Finally, the relations between the third row and the pure-gluino amplitude (fourth row) mirror

the situation found for the top of the diagram and are also of rank eight.

Combining all of the above into one large system of equations, the total rank of the super-

symmetric Ward identities pictured in figure 1 turns out to be 18. So, given any two of the 20

distinct amplitudes, one can calculate any other from this set employing the complete collection

of N = 1 SWI. The corresponding result has already been found by Grisaru and Pendleton in

the context of N = 1 supergravity [9], and recast recently in modern spinor-helicity form [56].

More explicitly, any two-gluino four-gluon amplitude Fi,I , with the gluinos situated at posi-

tions i and I, can be expressed in terms of the pure-gluon and pure-gluino amplitude as

Fi,I =4〈Ij〉[ij]〈g−g−g−g+g+g+〉 − εijk〈jk〉εIJK [JK]〈λ−λ−λ−λ+λ+λ+〉

−2∑

m,n∈i,j,k〈mn〉[nm], (4.18)

where i, j, k and I, J,K mark the set of negative and positive helicity particles respectively, and

the numerator contains implicit sums over j, k, J,K. For example,

F3,4 = 〈g−g−λ−λ+g+g+〉 =〈4|(1 + 2)|3]〈g−g−g−g+g+g+〉 + 〈12〉[56]〈λ−λ−λ−λ+λ+λ+〉

(k1 + k2 + k3)2.

(4.19)

A similar formula for all four-gluino two-gluon amplitudes can be found in the appendix of ref. [56].

20

4.5 The second N = 1 SUSY diamond

Recall [63] that the pure-gluon amplitude can be calculated from the latter three amplitudes

in eq. (4.13), namely

〈φ−φ−φ−φ+φ+φ+〉, 〈φ−φ−λ−λ+φ+φ+〉 and 〈φ−φ−g−g+φ+φ+〉 . (4.20)

The question that immediately arises is whether this set forms a basis for the complete set of

all six-point NMHV N = 2 amplitudes4 in helicity configuration X? We were not aware of a

direct answer to that question, so we took the following approach. As mentioned already in

subsection 4.2, we will consider a second set of six-point NMHV N = 1 supersymmetric Ward

identities, in addition to the N = 1 SWI for (g±, λ±) described in the previous subsection.

b

b

b

b b b b b b b b b

b b b b b b b b b

b b b b b b b b b

b b b b b b b b b

N = 2

N = 1

N = 1

〈g−g−g−g+g+g+〉〈g−g−λ−λ+g+g+〉〈g−λ−λ−λ+λ+g+〉〈λ−λ−λ−λ+λ+λ+〉〈φ−λ−λ−λ+λ+φ+〉〈φ−φ−λ−λ+φ+φ+〉〈φ−φ−φ−φ+φ+φ+〉

Figure 2: Amplitudes involving particles from a single N = 2 multiplet containing two N = 1

subsets.

In figure 2 the collection of six-point NMHV N = 2 amplitudes is depicted in helicity con-

figuration X. Every black dot denotes a particular amplitude. The top point represents the

pure-gluon amplitude 〈g−g−g−g+g+g+〉, the lowest point refers to the pure-scalar amplitude

〈φ−φ−φ−φ+φ+φ+〉, and the central point denotes the pure-gluino amplitude 〈λ−λ−λ−λ+λ+λ+〉.Supersymmetric Ward identities relate certain amplitudes from adjacent rows and the elements

of eq. (4.13) are encircled. The upper diamond-shaped region corresponds precisely to figure 1:

it is the subset of six-point NMHV N = 1 amplitudes built from the multiplet (g±, λ±) within

the N = 2 amplitudes. (There are additional states in the full N = 2 diamond in figure 2, of

course, even in the second row.)

However, the upper diamond-shaped region is not the only subset of six-point NMHV N = 2

amplitudes which can be related by N = 1 supersymmetric Ward identities. Stretching between

the pure-gluino and the pure-scalar amplitude there is a second region (referred to as the lower

diamond in the following), which satisfies relations similar to those in the upper N = 1 diamond.

4The term N = 2 amplitudes refers to all possible amplitudes that can be constructed exclusively from particles

from a single N = 2 multiplet and its CPT conjugate, (g±, λ±m, φ±) with m = 1, 2 [83].

21

The modified supersymmetry operator Q will now act on a multiplet consisting of scalars (φ+, φ−)

and gluinos (λ+, λ−) via

[

Q(η), φ+(p)]

= 〈pη〉λ+(p),[

Q(η), λ+(p)]

= − [pη]φ+(p),[

Q(η), φ−(p)]

= [pη]λ−(p),[

Q(η), λ−(p)]

= −〈pη〉φ−(p) , (4.21)

which can be easily derived by identifying the supercharges of N = 2 supersymmetry, Q1 and

Q2, with Q and Q respectively.

Writing down the set of supersymmetric Ward identities generated by acting with a super-

symmetry generator Q on the source term 〈φ−φ−φ−λ+φ+φ+〉, one encounters the same structure

derived in eq. (4.16):

[4η]〈φ−φ−φ−φ+φ+φ+〉 − [1η]〈λ−φ−φ−λ+φ+φ+〉− [2η]〈φ−λ−φ−λ+φ+φ+〉 − [3η]〈φ−φ−λ−λ+φ+φ+〉 = 0. (4.22)

In fact, one can show that the complete system of supersymmetric Ward identities and amplitudes

for the lower diamond, ranging from the pure-gluino to the pure-scalar amplitude, can be obtained

from the original N = 1 system considered in figure 1 by exchanging

Q ↔ Q

[ ] ↔ 〈 〉g+ ↔ φ+

g− ↔ φ−. (4.23)

This symmetry corresponds geometrically to reflecting figure 2 about a horizontal line passing

through the central point 〈λ−λ−λ−λ+λ+λ+〉.

The second system of supersymmetric Ward identities in the lower diamond is obviously of

the same rank as the original system. However, in contrast to the upper diamond it contains two

of the known amplitudes from ref. [63],

〈φ−φ−φ−φ+φ+φ+〉 and 〈φ−φ−λ−λ+φ+φ+〉 , (4.24)

which allows the calculation of any other amplitude in the lower N = 1 set. In particular, the

pure-gluino amplitude 〈λ−λ−λ−λ+λ+λ+〉 ( in figure 2), which is the element connecting the

upper and lower set of equations, can be determined. Having done so, there are now two known

amplitudes from the upper N = 1 diamond, the pure-gluino and the pure-gluon amplitude [63],

which in turn is the precondition for determining any amplitude from the upper N = 1 region.

22

In other words: any six-point NMHV amplitude in the two shaded regions in figure 2 can be

calculated from eq. (4.13).

In the next section, we will complete the ellipses on the left-hand side of the decomposi-

tion (4.12) by two gravitini and two gravitons, and KLT-factorize the result in such a way that

the desired six-point closed-string (N = 8 supergravity) amplitude can be related to a set of

two-gluino four-gluon N = 4 SYM amplitudes. The SYM amplitudes are available in turn by the

two-step procedure described above.

5 E7(7) symmetry for α′-corrected amplitudes?

As explained in the last section, the most accessible way of testing the double-soft scalar limit

relation is to calculate the N = 8 supergravity amplitude,

〈X1234 X1235 F 5+F−4 B+ B−〉 = KLT

[

〈g− λ4+ g+ λ−4 g+ g−〉L × 〈g+ λ−

5 λ5+ g− g+ g−〉R]

, (5.1)

a particular version of eq. (4.12). The determination of the right-hand side of eq. (5.1) will be

done by employing the two-step procedure described in the last subsection.

How should we obtain the pure-gluino amplitude 〈λ−λ−λ−λ+λ+λ+〉 from the amplitudes

in eq. (4.24) in the first step? An expression relating any six-point NMHV two-fermion four-

boson amplitude to the pure-fermion and pure-boson one has been given in eq. (4.18). We start

from eq. (4.19), employ the correspondence eq. (4.23) which transforms the pure-gluon amplitude

into the pure-scalar one, and solve the resulting equation for the pure-gluino amplitude:

〈λ−λ−λ−λ+λ+λ+〉 =(k1 + k2 + k3)

2〈φ−φ−λ−λ+φ+φ+〉 − 〈3|(1 + 2)|4]〈φ−φ−φ−φ+φ+φ+〉〈56〉[12] .

(5.2)

In the second step, we employ eq. (4.18) to obtain analytical expressions for all two-gluino four-

gluon amplitudes, allowing us to assemble finally the N = 8 amplitude.

In the same manner as explained in subsection 4.1 for the expansion to O(α′2) of a five-point

gravity amplitude, appropriate combinations of orders in α′ have to be added and permuted

on the right-hand side of eq. (5.1) in order to obtain the result including the R4 perturbation.

Explicitly, the third order in α′ can be obtained by evaluating

MO(α′3)6 =

−i

α′3s12s45

(

ASYM6 (1, 2, 3, 4, 5, 6)

×[

s35AO(α′3)6 (2, 1, 5, 3, 4, 6) + (s34 + s35)A

O(α′3)6 (2, 1, 5, 4, 3, 6)

]

+ AO(α′3)6 (1, 2, 3, 4, 5, 6)

×[

s35ASYM6 (2, 1, 5, 3, 4, 6) + (s34 + s35)A

SYM6 (2, 1, 5, 4, 3, 6)

]

)

+ P(2, 3, 4) . (5.3)

23

All amplitudes needed on the right-hand side of eq. (5.3) are two-gluino four-gluon amplitudes for

the helicity configurations X, Y or Z, which we have related by supersymmetry to the amplitudes

considered in ref. [63].

Before discussing the double-soft limit relation, we examine the single-soft limits, testing to

see whether the vanishing (2.4) observed in N = 8 supergravity still holds for the R4 matrix

elements. For the four-point amplitude, the factor of s1s2(s1 + s2) in the O(α′3) term in V(4)closed in

eq. (3.21) shows that the R4 matrix element vanishes at least as fast as the supergravity amplitude.

Similarly, using the forms (3.12) for the open string five- and six-point MHV amplitudes, together

with the appropriate KLT relations, we find numerically that the single-soft limit of the five- and

six-point MHV matrix elements of R4 vanish. That is, we construct a sequence of kinematical

configurations with the momentum of the scalar tending to zero, and we find that the R4 matrix

elements vanish. The vanishing is at the same rate as for the supergravity amplitudes, linearly

in the soft scalar momentum. (In the MHV case, it is sufficient to test the single-soft vanishing

for one particular amplitude containing scalars, because all other MHV amplitudes are related

by SWI involving ratios of spinor products that are constant in the soft limit.)

On the other hand, when we examine the single-soft limit of the non-MHV six-point R4 matrix

element (5.3) numerically, we find that it does not vanish.5 The question is whether this implies

the breaking of E7(7) symmetry by the R4 term. In principle there could be modifications to

the external scalar emission graphs that still allowed the symmetry to be intact (as happens in

the pion case). However, the R4 term does not produce any nonvanishing on-shell three-point

amplitudes. So it seems that the E7(7) symmetry is indeed broken, beginning at the level of the

non-MHV six-point amplitude.

One might wonder why the breaking shows up only at this level. If we consider the ten-

dimensional term e−2φt8t8R4 discussed in the introduction, which becomes e−6φt8t8R

4 after

transforming to Einstein frame, one might suspect a violation of the single-soft limit from the

non-derivative φ coupling already at the five-point level, expanding e−6φ = 1 − 6φ + . . ., and

with R4 producing two negative and two positive helicity gravitons. However, in four dimensions,

the dilaton belongs to the 70 of SU(8), while the gravitons are singlets, so a 〈φB−B−B+B+〉amplitude is forbidden by SU(8). Adding another scalar corresponds to providing a quadratic

SU(8)-invariant scalar prefactor for R4, and first affects NMHV six-point amplitudes.

Despite the apparent breaking of the E7(7) symmetry exhibited by the single-soft limit of the

NMHV six-point amplitude 〈X1234 X1235 F 5+F−4 B+ B−〉 at O(α′3), we now proceed to examine

the double-soft limits of this amplitude. First, though, we turn to the right-hand side of the

double-soft limit relation (2.5). Given the particular choice of amplitude (5.1), it is straightforward

to find an expression for the right-hand side. The operator

T 45 = ε12345

12354 ηi5∂ηi4 = − ηi5∂ηi4 (5.4)

5We thank Juan Maldacena for suggesting that we examine this limit, and for related discussions.

24

will act on the remnant of the six-point amplitude as

−6

∑

i=3

ηi5∂ηi4〈F 5+F−4 B+ B−〉

=

6∑

i=3

ηi5∂ηi4

(

∂

∂η35

)(

1

7!ε12345678

∂7

∂η41 . . . ∂η43∂η45 . . . ∂η48

)(

1

8!ε12345678

∂8

∂η61 . . . ∂η68

)

Ω4

= 〈F 4+F−4 B+ B−〉 − 〈F 5+F−

5 B+ B−〉 . (5.5)

Acting on particle 3, the operator changes the derivative with respect to η35 into a derivative

with respect to η34, thus effectively transforming the positive helicity gravitino F 5+ into F 4+.

Correspondingly, by acting on particle 4, again a derivative with respect to η45 will be changed

into one with respect to η44, this time transforming F−4 into F−

5 .

Restoring the kinematical weight factors in eq. (2.5), the final comparison will be made ac-

cording to the following formula:

〈X1234 X1235 F 5+F−4 B+ B−〉

∣

∣

∣

O(α′3)−→

1

2

[

p3 · (p2 − p1)

p3 · (p1 + p2)〈F 4+F−

4 B+ B−〉∣

∣

∣

O(α′3)− p4 · (p2 − p1)

p4 · (p1 + p2)〈F 5+F−

5 B+ B−〉∣

∣

∣

O(α′3)

]

. (5.6)

Given the complexity of the higher-order α′ corrections in the available amplitudes (see e.g.

eq. (3.18) at only O(α′2)), the analytical computation of the left-hand side of eq. (5.6) would be

very cumbersome. Instead the computation and comparison have been performed numerically

for a sufficient number of kinematical points.

For reference, we give numerical values at one sample double-soft kinematical point, with all

outgoing momenta fulfilling k2i = 0 and

∑6i=1 kµ

i = 0:

k1 = (−0.853702542142, +0.696134406758, −0.306157335124, +0.387907984368) × 10−4,

k2 = (+0.711159367201, −0.099704627834, −0.295472686856, +0.639142021830) × 10−4,

k3 = (+0.818866370407, +0.408234512914, −0.661447772542, −0.257630664418),

k4 = (−1.098195656456, −0.551965696904, −0.598319787466, +0.737143813124),

k5 = (−0.618073260483, +0.143671541012, +0.362410922160, −0.479615853707),

k6 = (+0.897416800850, +0.000000000000, +0.897416800850, +0.000000000000). (5.7)

At this point, with a particular external-state phase convention, the left- and right-hand sides of

the supergravity (O(α′0)) version of eq. (5.6) are given respectively by

−0.30572232 − i 0.89270274 ≈ −0.30615989 − i 0.89271337 , (5.8)

while the desired O(α′3) terms in eq. (5.6) are,

3.08397954 + i 9.00278816 ≈ 3.08775134 + i 9.00339016 . (5.9)

25

The difference between the left- and right-hand sides is due merely to the finite separation of the

point (5.7) from the double-soft limit. It can be made as small as desired by working closer to

the limit, using higher precision kinematics to avoid roundoff error.

The result is surprising: for any double-soft kinematical configuration considered, the left-

and the right-hand side of eq. (5.6) show complete agreement within numerical errors.

Given the available amplitudes from the two shaded regions in figure 2, one can perform

further tests for other N = 8 amplitudes. In addition to eq. (5.6), we have tested the double-soft

scalar limit for the following amplitudes

〈X1234 X1235 F 5+ F−4 F 4+ F−

4 〉∣

∣

∣

O(α′3)−→

1

2

[

+p3 · (p2 − p1)

p3 · (p1 + p2)〈F 4+F−

4 F 4+F−4 〉

∣

∣

∣

O(α′3)

− p4 · (p2 − p1)

p4 · (p1 + p2)〈F 5+F−

5 F 4+F−4 〉

∣

∣

∣

O(α′3)

− p6 · (p2 − p1)

p6 · (p1 + p2)〈F 5+F−

4 F 4+F−5 〉

∣

∣

∣

O(α′3)

]

(5.10)

and

〈X1234 X1235 X1235 X1235 X1235 X1234〉∣

∣

∣

O(α′3)−→

1

2

[

+p3 · (p2 − p1)

p3 · (p1 + p2)〈X1234 X1235 X1235 X1234〉

∣

∣

∣

O(α′3)

+p5 · (p2 − p1)

p5 · (p1 + p2)〈X1235 X1235 X1234 X1234〉

∣

∣

∣

O(α′3)

− p6 · (p2 − p1)

p6 · (p1 + p2)〈X1235 X1235 X1235 X1235〉

∣

∣

∣

O(α′3)

]

. (5.11)

Each limit shows complete agreement for any double-soft kinematical point.

6 Conclusion

Our computation shows that the double-soft limit of three distinct six-point O(α′3)-corrected

N = 8 matrix elements yields the corresponding weighted sum of four-point amplitudes, precisely

as dictated by E7(7) invariance [46]. However, this is quite puzzling, given the nonvanishing single-

soft limits of the same six-point amplitudes. The most likely possibility seems to be that the

double-soft limits will begin to fail, but only beginning with the NMHV seven-point amplitudes.

It would be very interesting to test this limit, but that is beyond the scope of the present paper.

Whether the three-loop cancellations [30, 31] can be explained by a simple symmetry argument

that originates in theE7(7)

SU(8) coset symmetry of N = 8 supergravity is still open. This work suggests

that the R4 term produced by tree-level string theory can be ruled out in this way, but other

26

dependences on scalars should be considered. The work of Green and Sethi [23] in ten dimensions

indicates that supersymmetry may forbid any R4 term, but an argument using supersymmetry

directly in four dimensions would be very welcome.

Of course there are higher-dimension potential counterterms than R4, which are relevant

beginning at five loops. It is possible that E7(7) and/or supersymmetry can be used to exclude

these counterterms as well, up to a certain dimension or loop order. However, at eight loops

a counterterm exists that is invariant under both supersymmetry and E7(7) [15, 54]. It is still

possible that E7(7) plays a more subtle role in the excellent ultraviolet behavior of the theory,

perhaps by relating somehow the coefficients of certain loop integrals making up the full multi-loop

amplitude.

Completely understanding the role of E7(7) will very likely be part of a fundamental explana-

tion of the conjectured finiteness of N = 8 supergravity. However, whether supersymmetry and

the coset symmetry alone are sufficient ingredients remains to be shown.

Acknowledgments

The authors would like to express their gratitude to Stephan Stieberger for conversations, and

to Tomasz Taylor and Stephan Stieberger for providing complete Mathematica expressions for

the NMHV six-point open-string amplitudes, which served as a starting point for this calculation.

Furthermore we would like to thank Zvi Bern, Michael Douglas, Henriette Elvang, Dan Freedman,

Renata Kallosh, Jared Kaplan, Hermann Nicolai, Stefan Theisen and especially Juan Maldacena

for helpful discussions. This research was supported by the U.S. Department of Energy under

contract DE-AC02-76SF00515. The work of J.B. was supported by the German-Israeli Project

cooperation (DIP) and the German-Israeli Foundation (GIF).

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